Pareto Distribution

(continuous probab dist for equations)

Usage:

ParetoDist (x, a, b)

Definition:

(a/b) (b/x) ^ (a + 1)

Parameters:

a is shape   b is location

Required:

a > 0    b > 0

Support:

x ⋝ b

Moments:

μ = a b / (a-1)  for a > 1   (∞ for a ⋜ 1)

σ^2 = a b^2 / [(a - 1)^2 (a - 2)]  for a > 2

(∞ if 1 <a ⋜ 2, non-existent if a ⋜ 1)

 γ1 = [2(1 + a)/(a - 3)] sqrt ((a - 2)/a)  for a > 3

β2 = 6(a^3 + a^2 – 6a - 2) / [a (a - 3) (k - 4)]  for a > 4

The Pareto distribution is a power law probability distribution found in a large number of real-world situations, such as the distribution of wealth among individuals, frequencies of words, size of particles, size of towns/cities, areas burnt in forest fires, size of some fractal features etc.  These are situations where there are many that are small and a few that are large (like the "Pareto principle", in which 20% of the population owns 80% of the wealth).

For any value of a, the distribution is "scale-free", which means that no matter what range of x one looks at, the proportion of small to large events is the same (i.e., the slope of the curve on any section of the log-log plot is the same).

Also known as the Bradford distribution.